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Is this Possible?

I was wondering about the comparison between the length of a line segment and arc length, but I have no idea how to construct it. What I am wondering is this: if you have an arc length of length "x", and you somehow get it to intersect (at two points) a line segment of length "x", where will they intersect? Here's an illustration of what I am thinking about:

So I am wondering how to construct this, if not, then what would happen if we could find the points of intersection between the arc and the line segment? If this helps, think about having a string pulled tight, and then tugging on the string. The string becomes arced, but not any bigger, so it should intersect in the same place, right?
Hopefully that is clear enough, thank you.